Question

1. $2x^3 - 16x^2 - 40x = 0$
2. $x^4 - 16x^2 = x^2 + 18$
3. $4x^4 + 35x^2 - 9 = 0$
4. $x^3 + 3x^2 = 24x + 72$
5. $2x^3 - 5x^2 + 40x - 100 = 0$
6. the population of a species is modeled by the equation $p(t) = -t^4 + 56t^2 + 240$, where $t$ is the number of years. find the approximate number of years until the species is extinct.

Explanation:

Step1: Factor out GCF

$2x(x^2 - 8x - 20) = 0$

Step2: Factor quadratic trinomial

$2x(x - 10)(x + 2) = 0$

Step3: Solve for x

Set each factor to 0:
$2x=0 \implies x=0$
$x-10=0 \implies x=10$
$x+2=0 \implies x=-2$

Step1: Rearrange to standard form

$x^4 - 17x^2 - 18 = 0$

Step2: Substitute $u=x^2$

$u^2 - 17u - 18 = 0$

Step3: Factor quadratic

$(u - 18)(u + 1) = 0$

Step4: Solve for u, substitute back

$u=18 \implies x^2=18 \implies x=\pm 3\sqrt{2}$
$u=-1 \implies x^2=-1 \implies x=\pm i$

Step1: Substitute $u=x^2$

$4u^2 + 35u - 9 = 0$

Step2: Factor quadratic

$(4u - 1)(u + 9) = 0$

Step3: Solve for u, substitute back

$4u-1=0 \implies u=\frac{1}{4} \implies x^2=\frac{1}{4} \implies x=\pm \frac{1}{2}$
$u+9=0 \implies u=-9 \implies x^2=-9 \implies x=\pm 3i$

Step1: Rearrange to standard form

$x^3 + 3x^2 - 24x - 72 = 0$

Step2: Factor by grouping

$x^2(x + 3) - 24(x + 3) = 0$

Step3: Factor out common binomial

$(x^2 - 24)(x + 3) = 0$

Step4: Solve for x

$x+3=0 \implies x=-3$
$x^2-24=0 \implies x=\pm 2\sqrt{6}$

Answer:

$x = -2, 0, 10$

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